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Chapter 3: Problem 46

Find the determinant of the elementary matrix. (Assume \(k \neq 0\).) $$\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & k & 1 \end{array}\right]$$

Short Answer

Expert verified
The determinant of the given elementary matrix is 1.

Step by step solution

01

Identify the type of matrix

Recognize that the matrix given is an elementary matrix derived from the identity matrix. It is obtained by multiplying the third row of the 3x3 identity matrix by k.
02

Calculate the determinant

For an elementary matrix, the determinant is the product of the elements of the main diagonal. In this case it is (1 * 1 * 1) = 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elementary Matrix
An elementary matrix is a matrix that results from performing a single elementary row operation on an identity matrix. Elementary row operations include swapping two rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another row.
  • These matrices are fundamental because they are used to simplify other matrices, especially when finding inverses.
  • In the context of the exercise, the elementary matrix is created by multiplying the third row of the identity matrix by a scalar, \( k \).

In practice, these matrices play an important role in understanding linear transformations and solving systems of linear equations. When you multiply an elementary matrix by any matrix, it performs the same row operation on that matrix.
Identity Matrix
The identity matrix is a special type of diagonal matrix with ones on the main diagonal, and zeros elsewhere. It acts like the number 1 in matrix algebra, meaning any square matrix multiplied by an identity matrix of compatible dimensions results in the original matrix.
  • For example, multiplying the identity matrix by itself or any other square matrix returns the same matrix.
  • Identity matrices are square, so an identity matrix can be of any size \( n \times n \), where \( n \) is the number of rows and columns.

In the constructed elementary matrix, the identity matrix was the starting point. By altering one row, a new elementary matrix was derived, showcasing how versatile identity matrices are when performing operations.
Main Diagonal
The main diagonal of a matrix comprises the elements that extend from the top left corner to the bottom right corner of the matrix. It is a significant concept because many matrix operations and properties are based on elements of this diagonal.
  • In a diagonal or an identity matrix, all non-diagonal elements are zero, simplifying calculations like determinants.
  • For square matrices, if all elements in the main diagonal are non-zero, the matrix likely has an inverse.

In the original exercise, the determinant was calculated using only the main diagonal elements. For an elementary matrix formed by multiplying a row of the identity matrix by a scalar, the determinant remains primarily influenced by the main diagonal components, which simplifies to the product of these diagonal values.

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Most popular questions from this chapter

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text (a) The characteristic equation of the matrix \(A=\left[\begin{array}{rr}2 & -1 \\\ 1 & 0\end{array}\right]\) yields eigenvalues \(\lambda_{1}=\lambda_{2}=1\) (b) The matrix \(A=\left[\begin{array}{rr}4 & -2 \\ -1 & 0\end{array}\right]\) has irrational eigenvalues \(\lambda_{1}=2+\sqrt{6}\) and \(\lambda_{2}=2-\sqrt{6}\)

The table below shows the numbers of subscribers \(y\) (in millions) of a cellular communications company in the United States for the years 2003 to \(2005 .\) (Source: U.S. Census Bureau)$$\begin{array}{l|c} \hline \text {Year} & \text {Subscribers} \\ \hline 2003 & 158.7 \\ 2004 & 182.1 \\ 2005 & 207.9 \\ \hline \end{array}$$ (a) Create a system of linear equations for the data to fit the curve \(y=a t^{2}+b t+c,\) where \(t\) is the year and \(t=3\) corresponds to \(2003,\) and \(y\) is the number of subscribers. (b) Use Cramer's Rule to solve your system. (c) Use a graphing utility to plot the data and graph your regression polynomial function. (d) Briefly describe how well the polynomial function fits the data.

Find the adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{rrr} 1 & 2 & 3 \\ 0 & 1 & -1 \\ 2 & 2 & 2 \end{array}\right]$$

Find an equation of the plane passing through the three points. $$(1,-2,1),(-1,-1,7),(2,-1,3)$$

Verify that \(\lambda_{i}\) is an eigenvalue of \(A\) and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. $$\begin{array}{l} A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array}\right] ; \quad \lambda_{1}=2, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right] \\ \lambda_{2}=0, \quad \mathbf{x}_{2}=\left[\begin{array}{r} -1 \\ 0 \\ 1 \end{array}\right] ; \quad \lambda_{3}=1, \quad \mathbf{x}_{3}=\left[\begin{array}{r} -1 \\ 1 \\ -1 \end{array}\right] \end{array}$$
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